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CURRENT SCIENCE, VOL. 102, NO. 11, 10 JUNE 2012 1539

*For correspondence. (e-mail: [email protected])

Computational fluid dynamics modelling ofsolid suspension in stirred tanks

Madhavi V. Sardeshpande and Vivek V. Ranade*Industrial Flow Modeling Group, Chemical Engineering and Process Development Division, National Chemical Laboratory, Pune 411 008, India

Solid suspension and mixing are crucial in manyimportant processes, including multiphase catalyticreactions, crystallization, precipitation, etc. In recentyears, various efforts have been made to numericallysimulate solidliquid flows in stirred vessels usingcomputational fluid dynamics (CFD). In this article,we present a brief account of our groups efforts atdeveloping and using CFD models for simulating solidsuspension in stirred tanks. Computational modelswere developed and evaluated by comparing modelpredictions with our data as well as published experi-mental data. A variety of experimental techniquesranging from torque and wall pressure fluctuations toultrasound velocity profiler were used. Efforts weremade to develop appropriate sub-models for captur-ing influence of the prevailing turbulence and solidvolume fraction on effective inter-phase couplingterms. A hysteresis in variation of the height of thecloud of suspended solid with impeller rotationalspeed was observed. The hysteresis, besides havingapplications in realizing better suspension at lower

effective power consumption, also offers an attractiveevaluation test for CFD models. A new way to carryout dynamic settling of solid cloud by sudden impellerstoppage has been developed. The approach, modelsand results presented here will be useful for extendingapplications of CFD models for simulating industrialstirred slurry reactors as well as further research inthe field.

Keywords: Computational fluid dynamics, hysteresis,

solid suspension, stirred tanks.

SOLIDsuspension and mixing are crucial in many impor-

tant processes, including multiphase catalytic reactions,crystallization, precipitation, leaching, dissolution, co-

agulation and water treatment. Stirred tanks are fre-

quently used for these purposes in chemical, biochemical

and mineral processing industries, because of their ability

to provide excellent mixing and contact between the solid

and liquid phases. Apart from providing a good suspen-

sion of solid, stirred vessels also offer excellent heat and

mass transfer. Despite their widespread use, the design

and operation of these tanks to ensure the desired quality

of suspension has remained a challenging problem over

the past decades. Suspension quality depends upon com-

plex interactions of impeller-generated flow, turbulence

and solid loading. Conventionally, solid suspension in

stirred reactors is characterized by impeller speed requi-

red for just off-bottom suspension. An important consid-

eration in the design and operation of slurry reactors is

the determination of the states of the solid suspension, at

which point no particles reside on the vessel bottom for a

long time. Such a determination is critical to enhance theperformance of the reactor, because until such a condition

is achieved, the catalyst is not effectively utilized.

Clearly, it is important to understand these processes and

the effect of hydrodynamics on them while designing

such reactors. The design of stirred tanks requires the

knowledge of the flow field (like velocity, turbulence

intensity, hold-up, distribution of the dispersed phase,

etc.) and the understanding of the effects of various sys-

tem parameters like impeller type, power input, number

of baffles, etc. on the desired process result.

Over the past few decades, solidliquid flows in stirred

vessels have been studied using numerical simulations

based on computational fluid dynamics (CFD) frame-work. Dispersed solidliquid flows can be modelled

using either the EulerianEulerian approach or the Eule-

rianLagrangian approach. The former approach uses the

concept of interpenetrating continua to formulate continu-

ity and momentum balances for each phase separately15.

Although CFD provides a platform that can be used to

obtain significant insights into complex multiphase flow

problems, it is necessary to validate the model predictions

extensively with experimental data before they can be

confidently used for the design and optimization of indus-

trial reactors. The experimental studies reported in the

literature mostly consist of axial measurement of concen-tration profiles in the vessel610, and ignore the radial

gradients that exist in the reactor. As a result, the major-

ity of CFD studies for solidliquid stirred tanks are either

devoted to the improved prediction of axial solid concen-

tration profiles1,4,10,1113, or are focused on the prediction

of particle suspension height in a stirred vessel3. The pre-

dictions for the solid flow and distribution in the tank

have not been extensively evaluated yet. Such evaluations

are necessary to facilitate further improvements and

applications of CFD models for the design and scale-up

of solidliquid stirred tank reactors. Overall, studies on

solid suspension in stirred tanks can be schematically

shown, as in Figure 1.

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CURRENT SCIENCE, VOL. 102, NO. 11, 10 JUNE 20121540

Figure 1. Studies of solid suspension in stirred tanks.

Figure 2. States of solid suspension. a, Partial suspension. b, Com-plete suspensions. c, Uniform suspension.

Critical analysis of the reported literature shows that

there is significant uncertainty in the estimation of the

inter-phase drag force on the solid particles in a turbulentfluid. The inter-phase drag force was found to affect the

suspension of solid particles. Several correlations relating

the inter-phase drag force with the solid volume fraction

and the prevailing turbulence are available4,1417

. How-

ever adequate guidelines to select appropriate models of

inter-phase drag force for the simulation of solidliquid

flows in the stirred reactors are not available. Therefore,

it is essential to carry out development of CFD models

for solidliquid flows in stirred vessels and evaluate the

model predictions by comparing with reliable experimen-

tal data.

This article summarizes the work done by our group inrecent years on the modelling of solid suspension in

stirred tanks and the evaluation of these models with

some conventional and a couple of new approaches. The

current status of CFD models and their ability to simulate

suspension quality, effective drag coefficient and liquidphase mixing are briefly discussed. Comparison of simu-

lated results with new experiments such as hysteresis in

cloud height and settling of solid at sudden impeller-

stoppage is then presented followed by a summary.

Suspension quality

The process of solid suspension can be broadly divided

into three regimes, i.e. on-bottom suspension regime, off-

bottom suspension regime (complete suspension) and

homogeneous (uniform) suspension regime18

, as shown in

Figure 2.Complete suspension impeller speed (Njs) is the speed

at which all particles are lifted up from bottom of the ves-

sel such that particles do not spend more than 12 s at the

bottom of the vessel19. Though the concept of Njs was

introduced 60 years ago19, it is still used as a primary

design parameter. Numerous correlations on just-

suspended speed for different operating and design con-

ditions have been published2022

. These are all similar to

the Zwietering correlation, except for variations in the

exponents of different terms. From a physical standpoint,

the state of suspension of solid particles in the reactor is

completely governed by the hydrodynamics and turbu-

lence prevailing in the reactor. The interaction of the

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Figure 3. Pictorial view of solid suspension.

Figure 4. Suspension quality with respect to impeller speed24.

particles with the liquid flow field and also the interac-

tions with other particles (significant for dense systems)

determine the motion of solid particles within the reactor.

Sardeshpande et al.23

focused on developing correlations

for just-suspension speed and carried out systematic

experimental study to characterize solid hydrodynamics

in slurry reactors using various experimental techniques.

Similarly, in a solidliquid system, one can visually

observe an interface that distinguishes two regions in thevessel: a region containing suspended solid (cloud

region) and a clear liquid region, once the quasi steady

state is achieved. At a particular impeller rotation, solid

get lifted to maximum height within the fluid forming an

interface between suspended solid and clear liquid. The

height of this interface from the tank bottom is called the

cloud height. This interface (i.e. cloud height) depends

on the existing fluid and particle characteristics, vessel

and impeller configuration and impeller speed. A picto-

rial view of cloud height in a stirred vessel is given in

Figure 3.

It is well known that the prevailing suspension quality

controls the liquid-phase mixing in the reactor. Therefore,

it becomes essential to first adequately predict the quality

of the suspension for different impeller rotational speeds.

Different criteria are available to characterize the suspen-

sion quality in the reactor. Kasat et al.24

used two criteria:

one based on the standard deviation of the solid concen-

tration (calculated using eq. (1)) and the other based on

the cloud height to qualitatively quantify the suspension

quality (Figure 4).

2

avg1

11 .

n

i

C

n C

=

=

(1)

For uniform (homogeneous) suspensions, the value of the

standard deviation is found to be smaller than 0.2

(< 0.2). However, for the just-suspension condition

the value of the standard deviation lies between 0.2 and

0.8 (0.2 < < 0.8), and for an incomplete suspension,

> 0.8.

Measurement of height of cloud within the stirred ves-

sel provides qualitative indication of suspension quality.

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Figure 5. States of solid suspension by Takenaka et al.28.

The literature provides extensive data on cloud height

measurements2528. Bittorf and Kresta29developed a cor-

relation for estimating cloud heights. Kraume25observed

that if cloud height reached up to 90% of tank height(which is equal to the tank diameter), the corresponding

impeller speed will be the same as that required for just

suspension speed,Njs. Kasat et al.24

compared suspension

quality using standard deviation and cloud height (Figure

4) and their results agreed with Kraumes criteria for just-

suspension speed.

Some discrepancies were observed in previously re-

ported studies regarding cloud height. For example, Hicks

et al.26showed monotonic behaviour of increasing cloud

height with increasing impeller speed. However, Bujalski

et al.27and Takenaka et al.28observed five stages of solid

suspension: (i) At lower impeller speed, small amounts of

particles are lifted up and distributed throughout the ves-sel. (ii) With an increase in the stirring speed, an interface

appears between the suspension and clear liquid layer in

the upper portion of the vessel. (iii) The height of the

interface reduces with stirring speed. (iv) Further increase

in stirring speed resulted in an increase in the height of

the interface and a decrease in the size of the upper clear

layer. (v) The interface between the cloud and clear liquid

layer disappears. The solid become fully suspended,

i.e. Njs is achieved, typically at stage (iv), as shown in

Figure 5. Our groups work on the influence of impeller

speed on cloud height is discussed in the next section.

Effective drag coefficient for solid particles instirred tank

During the last two decades, many studies have reported

CFD simulations (single as well as multiphase) of flow

field in a stirred reactor (see, for example, references

cited in Chapter 10 of Ranade30

). CFD models were

shown to be successful in simulating single-phase flow

generated by impeller(s) of any shape in complex reac-

tors30. Several attempts have also been made to simulate

the solidliquid flows in stirred reactors (see for example,

refs 4, 11 and 31). These efforts on developing CFD

models of solidliquid flows in a stirred tank mainly dif-

fer from each other with respect to how the model treats

impeller rotation, multiphase turbulence and effective

drag coefficient, with the last parameter being the most

crucial.Experimental measurements14,15

have demonstrated

that the prevailing bulk turbulence modifies the particle

drag coefficient to a substantial extent. In stirred tanks,

prevailing turbulence levels are controlled by impeller ro-

tation and therefore demand different correlations of ef-

fective drag coefficient than solidliquid flows without

an external turbulence controlling agency. Brucato et al.15

have developed a correlation for estimating effective drag

coefficient based on the experimental data collected using

the TaylorCoutte-type apparatus. They attempted to

correlate the ratio of effective drag coefficient in the pre-

sence of turbulence to effective drag coefficient of a

single particle settling in a quiescent liquid with the ratioof particle diameter to Kolmogorov length scale of turbu-

lence. The idea was reasonably successful and several

correlations based on this were developed (see Appendix

1 for a list of representative correlations). Khopkar et al.4

developed a computational meso-scale model of flow

over assembly of suspended particles with externally

adjustable prevailing turbulence levels. They have carried

out simulations based on a unit cell concept shown

schematically in Figure 6. Their simulations clearly indi-

cate that the effective drag coefficient depends on a ratio

of particle diameter to the Kolomogorov length scale. The

simulated values of effective drag coefficient were rea-sonably described by the functional form of a correlation

proposed by Brucato et al.15

, albeit with a different pro-

portionality constant (see Appendix 1).

We carried out several steady-state simulations based

on the effective drag coefficient developed from the unit

cell approach and observed reasonable agreement

between simulated and published experimental results

over a wide range of key parameters such as impeller

types, particle diameter, particle loading and size of the

stirred tank. The turbulent dispersion of dispersed/

suspended phase was modelled using the turbulent diffu-

sivity,D12(see eq. (1) in Appendix 1). It should be noted

that the contribution of turbulent dispersion force is

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Figure 6. Unit cell approach developed by Khopkar et al.4.

Figure 7. Comparative study of drag correlations using RT and PBTD by Khopkar et al.4.

significant only when the size of turbulent eddies is larger

than the particle size. In the case of a solidliquid stirred

reactor, even for the laboratory scale, the ratio of the

largest energy containing eddy (in mm) and the particle

size was found to be around 10. Therefore, the contribu-

tion of the turbulent dispersion is likely to be significant.

The previously reported numerical studies have also high-

lighted the importance of the modelling of turbulent dis-persion force while simulating solid suspension in a stirred

reactor3234

. Considering these results, the turbulent dis-

persion of the dispersed phase was considered in the pre-

sent study. Also, the default value of the dispersion

Prandtl number, 0.75, has been used here (the model

equations used for these simulations are listed in Appen-

dix 1). A sample of the comparisons of simulated results

with the experimental data of Yamazaki et al.6, and

GodFrey and Zhu

9

was presented by Khopkar et al.

4

, asshown in Figure 7. It can be seen that the developed

2

4p

s

d

l

= dp Diameter of cylindrical object

s Solid hold-up

Grid details:

Total grid 199,809

xy 447 447

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Figure 8. Computational grid and solution domain24.

Figure 9. Mixing time at various locations in solution domain24.

computational model describes the experimental data

adequately.

Liquid-phase mixing

It is important to understand liquid-phase mixing in the

presence of solid suspension. Kasat et al.24

developed a

comprehensive CFD model to gain insight into solid sus-pension and its implications on the liquid-phase mixing

process in a solidliquid stirred reactor. They used the

approach proposed by Khopkar et al.4to simulate the tur-

bulent solidliquid flows in a stirred reactor. The model

predictions were compared with the experimental data of

axial solid concentration profile reported by Yamazaki et

al.6. The validated model was further extended to simu-

late the liquid-phase mixing. Completely converged flow

results were used for simulating the liquid-phase mixing,

assuming that the addition of tracer does not influence the

fluid dynamics in the stirred reactor. The mixing time is

defined as the time required to achieve the specifieddegree of homogeneity from the time at which the tracer

is added to the reactor. The species transport equationwas solved till the desired mixing was achieved. The

results indicated that adequate care needs to be taken to

ensure that the simulated results are not sensitive to the

time-step used. Typically low time-step of about 0.001 s

needs to be used with an adequate number of internal

iterations per time-step to ensure acceptable convergence

at each time-step (~20 internal iterations per time-step).

The tracer history recorded at different locations distri-

buted over the reactor (Figure 8) was used to estimate

mixing time for a specified degree of homogeneity. The

maximum value of the mixing time obtained from all the

right different locations was considered as the effective

mixing time of the system.

The simulated results indicated that mixing time values

estimated from locations above the cloud and within the

solid cloud showed significant difference. The simulated

variation in mixing time values at four locations (1, 3, 5

and 6) two are in the fastest mixing region (near the

impeller discharge stream) and two are in slowest mixing

region (near top surface) for all the operating conditions

is shown in Figure 9. The mixing time increases with

increase in the impeller rotational speed, reaches a maxi-

mum and then drops gradually with further increase in the

impeller rotational speed till the just off-bottom suspen-

sion condition is reached. The mixing time was thenfound to remain constant till the system approaches a

complete suspension condition and then the mixing time

slowly reduced with further increase in the impeller rota-

tional speed. The computational model has predicted

maxima in the mixing time at one-third of the complete

suspension speed of the impeller. The maximum mixing

time was found to be almost 10 times the minimum value

of the mixing time obtained at N = 40 rps. It can be seen

from Figure 9 that the difference between the predicted

values of mixing time near the impeller discharge stream

and near the top surface is dependent on the overall sus-

pension quality in the reactor. It was observed that forincomplete as well as for just off-bottom suspension

Grid details:

rz : 57 93 57

Impeller blade : 14 1 14

Inner region : 8 k47j39

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regimes, the mixing time values in clear liquid layer were

significantly higher (22.5 times) than those in the slurry

region (within the cloud of suspended solid). However,

the difference between the mixing time values of all the

four locations decreased after the start of the complete

suspension regime. The delayed mixing occurring in thetop clear liquid is responsible for the significantly larger

mixing time observed. The presence of very low liquid

velocities in the top clear liquid layer is responsible for

the delayed mixing.

It will be useful to briefly comment on the influence of

impeller shape and location on liquid-phase mixing.

These aspects have been extensively studied earlier23,35

.

These, and many other studies, clearly indicate that the

liquid-phase mixing time is essentially determined by the

mean flow generated by the impeller. Therefore, the

dimensionless mixing time can be estimated as:

3

mix1

,Q

TN

N D

where N is the impeller speed (rps), NQ the pumping

number of the impeller, mixthe mixing time, Tthe vessel

diameter and D is the impeller diameter. Pumping num-

bers of many of the commonly used impellers like Rush-

ton turbine, pitched-blade turbine and hydrofoils are not

very different from each other and therefore dimen-

sionless mixing time for these impellers is more or less

the same (though power consumption will be signifi-

cantly different).

Hysteresis in cloud height

The results discussed so far essentially use steady state

experimental data and simulations for understanding

solid suspension. Recently, Sardeshpande et al.36 ex-

plored the possible use of unsteady experiments and

simulations for better understanding of solidliquid flow

in stirred tanks and for evaluation of CFD models. Ex-

periments were conducted to visualize cloud height inside

the stirred vessel, similar to those used by Takenaka etal.

28. Cloud height depends on fluid and particle charac-

teristics, vessel and impeller configuration, and impeller

speed. At impeller speed of 150 rpm, it was observed that

large amounts of solid particles settled at the bottom of

the vessel. Only a small fraction of solid particles was

suspended in the liquid. The effective suspended solid

loading was, therefore, quite low, and solid were suspen-

ded up to higher levels resulting in higher cloud height

(high axial gradient in solid hold-up). When impeller

speed increased to 220350 rpm, flow stream from the

impeller impinged on the settled solid bed leading to

the suspension of more solid particles. This resulted inthe suspension of higher solid volume fraction from the

bottom of the vessel affecting effective slurry density and

flow near the impeller and lower cloud height than that

observed at 150 rpm. With further increase in the impel-

ler speed beyond 220 rpm, the bed region below the

impeller which reduces the strength of impeller flow

stream vanished, leading to unrestricted flow resulting inhigher cloud height. At 445 rpm, most of solid were more

or less uniformly suspended. Therefore, further increase

in impeller speed does not affect the cloud height. Thus,

the non-monotonic cloud height behaviour reported by

Sardeshpande et al.36

was in agreement with that of

Takenaka et al.28

for impeller PBTD-6. With this back-

ground, hysteresis in cloud height was studied.

Preliminary visual observations of cloud height indi-

cated that if different paths are followed in the operating

conditions, such as increase in the impeller speed from

lower to higher rpm, and vice versa then there is a possi-

bility of hysteresis in cloud height (Figure 10).At higher solid loadings, more energy is required to

suspend particles from the bottom of a vessel and to keep

them in suspended condition, compared to the energy re-

quired to prevent the already suspended particles from

settling. Typical experimentally observed hysteresis in

cloud height is shown in Figure 11a. It is clear from the

data that the observed cloud height at a specific impeller

speed is different when it is reached by either increasing

or decreasing the impeller speed. The observed hystere-

sis in cloud height was dependant on impeller type,

speed and solid loading. The hysteresis was observed

because of differences in slurry density near the impeller

and the formation of recirculation zones while increasing

or decreasing the impeller speed.

The observed differences in cloud height may be due to

the unsteady nature of solid suspension. To ensure that

the observed hysteresis is not just a transient effect,

experiments were carried out for a amount of substantial

time. The time history of cloud height measurements is

shown in Figure 11b. It was observed that with changes

in operating conditions, there was a difference in the

cloud height for the same impeller speed even after 30 min.

Figure 10. Cloud height using two different paths.

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Figure 11. a, Hysteresis in cloud height with PBTD-6 at 7% v/v. b, Cloud height behaviour at different timescales.

Figure 12. Contour plots of cloud height. a, Two initial conditions. b, Five radial planes from impeller to wall.

Efforts were made to capture the observed hysteresis in

cloud height using the developed CFD models. Simula-

tions were carried out at four different impeller speeds(220, 275, 350 and 450 rpm). For each impeller speed,

single-phase simulations were carried out and steady-

state convergence was achieved, where complete devel-

opment of axial flow pattern was observed. Simulations

were then carried out with two different initial condi-

tions: one with completely suspended solid by patching

mean solid volume fraction in the solution domain (i.e.

uniform patching) and the second with specified solid

corresponding to the mean solid volume fraction as set-

tled on the bed with maximum packing volume fraction

(i.e. bottom patching). This was done by adopting two

different approaches, and steady state and unsteady state

simulations were carried out using drag correlations of

Brucato et al.15and Khopkar et al.4. The contour plots of

solid volume fractions with two initial conditions using

the correlation of Brucato et al.15

are shown in Figure 12.Funnelling of solid explained the different axial and

radial levels, i.e. uneven solid distribution inside the

vessel. Therefore, an attempt was made here to adopt

circumferential axial averages of solid volume fraction at

five different radial locations (Figure 12).

Different levels of solid volume fractions at five radial

locations were observed. Averaged values of solid vol-

ume fraction from solid-rich region to clear liquid region

helped quantify cloud height. A comparative study

showed that the correlation of Brucato et al.15

was over-

predicting the results, whereas that of Khopkar et al.4

predicted experimental data of hysteresis for PBTD-6 rea-

sonably well, as shown in Figure 13.

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Figure 13. Prediction of hysteresis in cloud height using the correla-

tion of Khopkar et al.4

.

Figure 14. Schematic view for the dynamic settling of a cloud ofparticles.

Figure 15. Sample plot of solid settling time using PBTD-6.

Impeller location plays an important role in cloud

height as well as hysteresis in cloud height. Generally

lower impeller clearance (from the bottom) results in a

better distribution of solid. Numerous data and reports on

the influence of impeller location on solid suspension are

available on cloud height26,27,37,38. These results are notdiscussed here for the sake of brevity and the cited papers

may be referred for more information.

Dynamic settling of a cloud of particles atsudden impeller stoppage

Dynamic settling experiments were carried out by sud-

denly stopping the impeller when the particles were in

suspended condition. The settling of the suspended cloud

of particles was then monitored. The effective settling

velocity of the cloud particles can provide useful

information about the solid suspension in stirred tanks.

The schematic of the experimental set-up is shown in

Figure 14.

Dynamic settling of suspended solid was characterized

by using a high-speed camera. The height of solid bed

being accumulated on the side wall of the vessel due to

settling was recorded at a speed of 30 frames/s for a dura-

tion of 60 s (with the camera at a fixed position). A

black-coloured cloth was used as a background and illu-

mination (using halogen lamp) was arranged from the

front. For every experiment, the impeller was rotated at a

specific speed for at least 10 min to ensure that a quasi-

steady state of solid suspension corresponding to thatimpeller speed is achieved. The impeller speed was var-

ied in the 220445 rpm range. After achieving the quasi-

steady state of solid suspension, the power to the impeller

was abruptly switched-off. It was observed that the im-

peller stopped completely within 1 s after switching-off

the power. Images of settled bed and the bed height were

recorded at the vessel wall. It should be noted that the

bed settled at the vessel bottom is radially non-uniform.

The bed height at the vessel wall may therefore change

with impeller speed, even for the same volume fraction of

solid in the vessel. Despite a non-uniform bed, the

experimental methodology used in this study is ade-quately accurate for identifying solid settling time (since

it depends on the temporal profile of the bed height at the

vessel wall and not on its absolute value). Reproducibility

of settling of the cloud of particles was verified by

repeating all experiments at least three times. A sample

plot of determination of settling time of the cloud of

particles is shown in Figure 15.

These settling experiments were simulated using the

CFD models. The results obtained for solid volume frac-

tion of 7% v/v and particle size of 250 m in the impellerspeed range 220445 rpm are discussed below.

After the converged pseudo-steady state simulation of

solid suspension in stirred tanks, the impeller rotational

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Figure 16. a, Transient simulations of settling of cloud at 350 rpm. b, Settling time at 350 rpm and 445 rpm.

Figure 17. Comparative study of settling velocity of cloud experi-ments and CFD modelling.

speed was suddenly set to zero to simulate sudden switch-

off of the impeller. In reality, the impeller takes about 1 s

to stop rotating. In order to simulate this, the impeller

speed was linearly decreased from the initial value to

zero within 1 s and thereafter kept at no rotation condi-

tion. The simulated results for these two different ways ofcapturing the sudden switching off of the impeller did not

show significant differences. Subsequent simulations

were therefore carried out by suddenly setting the impel-

ler rotational speed as zero. Unsteady-state simulations

were carried out after setting this condition and a volume-

weighted average of key parameters, including mean

velocity of water, mean velocity of solid particles and

turbulent dissipation rate were monitored/recorded as a

function of time after the sudden switching off. A sample

of the results is shown in Figure 16.

It should be noted that when the impeller is not

switched-off, flow in the stirred tank is fully turbulent.After setting the impeller speed to zero, gradually the tur-

bulence decays and eventually the liquid becomes still.

The rigorous model of decaying turbulence is quite com-

plex. Fortunately, it is not necessary to model the decay-

ing turbulence rigorously here. Instead, two asymptotes

of fully developed turbulence and laminar flow (once the

volume-averaged turbulence kinetic energy reduced

below a certain threshold) give reasonable simulations. A

sample of simulated results of settling velocity of the

cloud of suspended solid is shown in Figure 17 along

with the experimental data. Details of the model equa-

tions and effective drag coefficient correlation are dis-

cussed in Sardeshpande et al.36. The settling velocity of

the cloud was calculated from a ratio of initial cloudheight and solid settling time. The solid settling time was

defined based on the decaying velocities of the solid par-

ticles. The settling time was defined as the time at which

the volume-averaged solid particle velocity becomes less

than 1% of the tip speed (tip speed before stopping the

impeller). This was found to agree with the experimental

data of settling time of the cloud of particles.

CFD simulations using the model of Sardeshpande et

al.33were found to capture the key characteristics of solid

suspension in stirred tank reasonably well.

Summary and path forward

Here we have presented a brief account of our groups

efforts in developing and using CFD models for simulat-

ing solid suspension in stirred tanks. Hysteresis in cloud

heights of suspended solid in stirred tanks and experi-

mental data with sudden impeller-stoppage have been

reported. The key conclusions based on this work are as

follows:

Cloud height of suspended solid in stirred tanks showsnon-monotonic behaviour with impeller rotational

speed. Contrary to the simple expectation, cloudheight was found to decrease with increase in impeller

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rotational speed, till the latter crossed a certain critical

value. Beyond this value, cloud height increased with

impeller rotational speed. This behaviour occurred

because of the complex interactions of the impeller

stream and solid bed settled on the vessel bottom, and

the influence of slurry density on impeller pumping. Cloud height of suspended solid exhibits hysteresis

with respect to impeller rotational speed. It was obser-

ved that cloud height at a specific impeller speed, if

approached by reducing the impeller speed, is gener-

ally higher than if approached by increasing the im-

peller speed. This observation may be used to make

better solid suspension at a specific impeller speed by

initially running the impeller at a higher rotational

speed and then gradually reducing it to the specific

target impeller speed.

The mixing time was found to increase with impeller

rotational speed initially till it reached a maximumvalue and then dropped gradually with further increase

in the impeller rotational speed. Once the just off-

bottom suspension was achieved, the mixing time

remained more or less constant till the system appro-

ached complete (uniform) suspension of solid. Be-

yond this stage, the mixing time slowly decreased

with further increase in impeller rotational speed.

A unit cell approach was useful in understanding theinfluence of the prevailing turbulence, solid volume

fraction and particle Reynolds number on the effective

drag coefficient. The influence of prevailing turbu-

lence on the drag coefficient can be estimated by the

functional form proposed by Brucato et al.15. The

results obtained from the unit-cell approach and sub-

sequent applications to solid suspension in stirred

tanks indicate that for higher solid loading and larger

particle Reynolds numbers, the proportionality con-

stant appearing in the correlation of Brucato et al.15

needs to be reduced.

CFD models along with appropriately modified corre-lation for estimating the effective drag coefficient

were found to simulate solid suspension in stirred tanks

reasonably well using the multiple reference frame

approach and the standard kturbulence model. The

models were able to capture axial distribution ofsolid over a wide range of particle diameter, solid

hold-up and tank size.

The CFD models were also able to capture the obser-ved hysteresis and dynamic settling of solid ade-

quately despite some of the well-recognized issues

associated with the models.

It should be noted that the models used in this work

were primarily based on the EulerianEulerian approach

and two-equation turbulence models. In recent years,

attempts at designing several more complex models based

on large eddy simulations, lattice Boltzamnn simulationsor direct numerical simulations have been made for simu-

lating solid suspension in stirred tanks (see for example

Derksen39,40 and references cited therein). Though these

recent models look theoretically more appealing, the

model selection for solving industrial problems is often

dictated by the cost-to-benefit ratio in practice. At this

point of time, the approach based on multiple referenceframe and two-equation turbulence models often provides

adequate guidelines for solving industrial problems. Bet-

ter-quality experimental data are needed to improve such

models as well as to identify limitations of their applica-

bility. The better quality experimental data and identified

limitations of the current models will also provide re-

newed impetus for developing better models. Our group

is currently working on establishing local solid measure-

ment facilities as well as electric capacitance/resistivity-

based tomography systems. These facilit ies will be used

to archive new and better-quality experimental data. The

models and results presented here can be further extendedto enhance their application potential.

Nomenclature

N Impeller speed (rps)

Njs Critical impeller speed at just suspension (rps)

Di Impeller diameter (m)

CD Drag coefficient in turbulent liquid

CD0 Drag coefficient in still liquid

D12 Turbulent diffusivity (m/s)

dp Particle diameter (m)

12,iF

Interphase drag force (N)FDF Turbulent dispersion force

H Height of tank

pl Liquid pressure (N/m2)

ps Solid pressure (N/m2)

Qp Pumping capacity of impeller (m3/s)

R Radius of impeller (m)

Rep Reynolds number of particle

Reimp Reynolds number of impeller

g Acceleration due to gravity (m/s)

z Axial coordinate (m)

Greek letters

l Density of water (kg/m3)

s Density of solid (kg/m3)

m Density of mixture (kg/m3)

Turbulent kinetic energy (m2/s

3)

Cycle (circulation) time (s)

mix Mixing time (s)

Turbulent dissipation rate (m2/s

3)

Kolmogorov length scale (m)

Viscosity (kg/ms)

tm Turbulentviscosity of mixture (kg/ms)

tq Viscosity of turbulence (kg/ms) Shear stress (N/m)

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Subscripts

1 Liquid

2 Solid

l Liquid

s Solidq Phase number

t Turbulent

Appendix 1. Model equations

(A) The continuity equation for phase q without mass

transfer between phases is written as

, 12

( ).( ) 0.

q qq q q i q qU D

t

+ =

(1)

Here , , U are the volume fraction, density and mean

velocity of phase qrespectively. The momentum balances

for the liquid and solid phases respectively are

( ) ( )

( ) ( )

,, ,

( ) ( ), ,

12, , ,

( ) ( )

, ,

12,

( ).( )

. .

.( ) ( )

. .

.

l l l il l l i l i

lam t l l l l l il ij l ij

i s s s i s s s i

lam tur

s s s s ss ij s ij

s s i i i DF

UU U

t

p g

F U Ut

p p

g F F F

+

= +

+ +

=

+ + +

(2)

The term Fi represents the Coriolis and centrifugal forces

applied in the rotating reference frame and is written as

2 ( ).i i i i i iF U r =

(3)

In this work, Boussinesqs eddy viscosity hypothesis has

been used to relate the Reynolds stresses with gradients

of time-averaged velocity as

( ), , ,,

2(( ( )) ) ( ( )).

3

tur Ttq q i q i q iq ij U U I U = +

(4)

The standard k turbulence model with mixture proper-

ties was used.

,..( ) ( ) ,tmm m m i

m

U St

+ = +

1 2, [ ].k m mS G S C G C k

= = (5)

22

, ,1

( ( ) ) , .2

T mtm m i m i tm

C kG U U

= + =

(6)

1

1

1

.

n

q q qnq

m q q m nq

q q

q

U

U

=

=

=

= =

(7)

Standard values of the k model parameters have been

used in the present simulations (C1= 1.44, C2= 1.92,

C= 0.09, k= 1.0 and = 1.3).

The drag force termed as

2 0.51 2 1 2, 1, 2, 1,

12,

3 ( ( ) ) ( ).

4

D i i i i

i p

C U U U U F

d

= (8)

0

0

3

.D D p

D

C C dK

C

=

(9)

(B) List of drag correlations.

Schiller Naumann41

,0

0.68724 (1 0.15 )D pp

C ReRe

= +

Magelli14,16

0.4 tanh 1 0.6s

t p

U

U d

= +

Brucato et al.15,

0

3

41 8.76 10pD

D

dC

C

= +

Pinelli et al.16

,16

0.4 tanh 1 0.6s

t p

U

U d

= +

Khopkar et al.4,

0

351 8.76 10

pD

D

dC

C

= +

Fajner et al.17,

0.5

0.32 tanh 19 1 0.6p ls

t p l

U

U d

= +

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Received 14 November 2011; revised accepted 13 April 2012

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